L09 - Examples Example Consider the IVP 2y 3xy 3y = 5x 2x y(1 = 1 y(1 = 1(N H Strategy nd a FS for the associated homogeneous SOLODE 2x2y 3xy 3y =

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Examples Example: Consider the IVP 2 x 2 y 00 + 3 xy 0 - 3 y = 5 x y (1) = 1 ,y 0 (1) = 1 . ( NH ) Strategy: find a FS for the associated ho- mogeneous SOLODE 2 x 2 y 00 + 3 xy 0 - 3 y = 0 , ( H ) then, using Variation of Parameters, find a particular solution Y ( x ) of ( NH ) , which will give us the general solution y ( x ) = Y ( x ) + c 1 y 1 ( x ) + c 2 y 2 ( x ) , ( G ) then accommodate the Initial Conditions. See that ( H ) has the solution y 1 ( x ) = x .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Panic! I have forgotten how Reduction of Order works. Actually, I do remember that it uses Abel’s theorem: there is a second solution y 2 such that W [ y 1 ,y 2 ]( x ) = e - R p ( x ) dx . Now y 1 y 0 2 - y 2 y 0 1 = e - R 3 / 2 x dx = x - 3 / 2 = x - 3 / 2 = y 1 y 0 2 - y 2 y 0 1 y 2 1 = x - 3 / 2 x 2 = x - 7 / 2 = ± y 2 y 1 ² 0 = x - 7 / 2 = y 2 y 1 = Z x - 7 / 2 dx = - 2 5 x - 5 / 2 2
Background image of page 2
= y 2 ( x ) = - 5 2 x - 3 / 2 Thus { x,x - 3 / 2 } is a FS for ( H ) . It has Wronskian W [ y 1 ,y 2 ] = - 5 2 x - 3 / 2 . Now find a particular solution Y of ( NH ) . Panic! I have forgotten how Variation of Parameters works. Actually, I remember that it means looking for a particular so- lution of the form Y ( x ) = v 1 ( x ) y 1 ( x ) + v 2 ( x ) y 2 ( x ) with variable parameters v 1 ,v 2 . So I take some scratch paper and derive it right quick: 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Y 00 = v 1 y 00 1 + v 2 y 00 2 + v 0 1 y 0 1 + v 0 2 y 0 2 p Y 0 = v 1 p y 0 1 + v 2 p y 0 2 + p [ v 0 1 y 1 + v 0 2 y 2 | {z } =0 ] q Y = v 1 q y 1 + v 2 q y 2 L [ Y ] = v 1 L [ y 1 ] + v 2 L [ y 2 ] | {z } =0 + v 0 1 y 0 1 + v 0 2 y 0 2 | {z } = g results in the two linear equations v 0 1 y 1 + v 0 2 y 2 = 0 , v 0 1 y 0 1 + v 0 2 y 0 2 = g . for v 0 1 ,v 0 2 . They have the solutions v 0 1 = - y 2 g W [ y 1 ,y 2 ] and v 0 2 = y 1 g W [ y 1 ,y 2 ] .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/29/2009 for the course M 427K taught by Professor Fonken during the Fall '08 term at University of Texas at Austin.

Page1 / 18

L09 - Examples Example Consider the IVP 2y 3xy 3y = 5x 2x y(1 = 1 y(1 = 1(N H Strategy nd a FS for the associated homogeneous SOLODE 2x2y 3xy 3y =

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online